Your final decision sounds like the appropriate one. As you may know, sometimes a table can act as both a fact and dimension. It all depends upon the business questions the users want to answer--just be careful with the bridge table if you are going to use Analysis Services with this model.

BIDSHELPER and this whitepaper can assist you with this should your solution require an Analysis Services layer:

http://www.microsoft.com/en-us/download/details.aspx?id=137

http://bidshelper.codeplex.com/wikipage?title=Many-to-Many%20Matrix%20Compression